Question 1198673: Use a system of equations to solve the following problem
The sum of 3 integers is 314. The sum of the first and second integers exceeds the third by 40. The third integer is 26 less than the first. Find the three integers.
Answer by math_tutor2020(3820)   (Show Source):
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x = first integer
y = second integer
z = third integer- Fact 1: The sum of 3 integers is 314.
- Fact 2: The sum of the first and second integers exceeds the third by 40.
- Fact 3: The third integer is 26 less than the first.
x+y+z = 314 from fact 1
x+y = z+40 from fact 2
z = x-26 from fact 3
Plug z = x-26 into the first equation
Then solve for y
x+y+z = 314
x+y+x-26 = 314
2x+y = 340
y = -2x+340
Let's plug those y and z expressions into the second equation and solve for x.
x+y = z+40
x+(-2x+340) = x-26+40
-x+340 = x+14
-x-x = 14-340
-2x = -326
x = -326/(-2)
x = 163
Use that to find the second integer
y = -2x+340
y = -2*163+340
y = 14
Use that to find the third integer
z = x-26
z = 163-26
z = 137
Summary:
x = 163
y = 14
z = 137
Check:- x+y+z = 163+14+137 = 314 which verifies fact 1
- x+y = 163+14 = 177 which is 40 larger than z = 137, since 137+40 = 177, confirming fact 2
- 137 is 26 less than 163 since 163-26 = 137, which confirms fact 3.
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